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Questions and Answers
What is a linear combination of vectors?
What is a linear combination of vectors?
What is a spanning set of a vector space?
What is a spanning set of a vector space?
A set where every vector in the space can be represented as a linear combination of its vectors.
Define the span of a set.
Define the span of a set.
The set of all linear combinations of the vectors in the set.
Span(S) is a subspace of V.
Span(S) is a subspace of V.
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When is a set of vectors called linearly independent?
When is a set of vectors called linearly independent?
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A set S is called linearly dependent if it has _____ solutions.
A set S is called linearly dependent if it has _____ solutions.
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What are the steps to test for linear independence?
What are the steps to test for linear independence?
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A set is linearly independent if one of the vectors is a linear combination of others.
A set is linearly independent if one of the vectors is a linear combination of others.
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Two vectors in a vector space are linearly independent if one is a scalar multiple of the other.
Two vectors in a vector space are linearly independent if one is a scalar multiple of the other.
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Study Notes
Linear Combinations
- A vector v in a vector space V is a linear combination of vectors u1, u2, ..., uk if it can be expressed as c1u1 + c2u2 + ... + ckuk, where c1, c2, ..., ck are scalars.
Spanning Sets
- A subset S = {v1, v2, ..., vk} of a vector space V is a spanning set if every vector in V can be expressed as a linear combination of vectors in S.
- If S spans V, it means that the entirety of V can be reached by taking linear combinations of vectors in S.
Span of a Set
- The span of a set of vectors S = {v1, v2, ..., vk} is defined as the collection of all linear combinations of the vectors in S.
- Denoted as span(S) or span{v1, v2, ..., vk}, it represents all possible linear combinations using real numbers as coefficients.
- If span(S) = V, then it is stated that V is spanned by S.
Subspace Properties
- Span(S) is a subspace of the vector space V that includes all linear combinations of vectors in S.
- It is the smallest subspace containing S, meaning any other subspace that contains S must also include span(S).
Linear Dependence and Independence
- A set of vectors S = {v1, v2, ..., vk} is linearly independent if the equation c1v1 + c2v2 + ... + ckvk = 0 only holds for the trivial solution c1 = 0, c2 = 0, ..., ck = 0.
- If there are nontrivial solutions (other than the trivial one), then the set S is linearly dependent.
Testing Dependence/Independence
- To test whether a set S is linearly independent or dependent:
- Set up the equation c1v1 + c2v2 + ... + ckvk = 0 and transform it into a system of linear equations.
- Apply Gaussian elimination to assess the uniqueness of the solution.
- If the solution is unique (only trivial), S is independent; if there are nontrivial solutions, it is dependent.
Theorem on Linear Independence
- A set of vectors S = {v1, v2, ..., vk}, with k ≥ 2, is linearly independent only if at least one vector can be expressed as a linear combination of the others.
Corollary on Two Vectors
- Two vectors u and v in a vector space V are linearly independent if and only if neither is a scalar multiple of the other.
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Description
Test your knowledge on spanning sets and linear independence with these flashcards. Understand key concepts like linear combinations and spanning sets in vector spaces. Perfect for students studying linear algebra.
